show/hide this revision's text 3 clarification of my intent

I was studying the axioms of a category, and noted that one axiom says there is an element $1_X\in Hom(X,X)$ for any object $X$ which serves as the identity. Why is this axiom necessary? What happens if I drop this axiom?

Background: I can define the category of affine holomorphic symplectic varieties, by saying

  1. The objects are semisimple algebraic groups
  2. The morphisms $Hom(G,G')$ are affine holomorphic symplectic varieties with Hamiltonian $G\times G'$ action
  3. The composition of two morphisms $X\in Hom(G,G')$ and $Y\in Hom(G',G'')$ is given by the holomorphic symplectic quotient $X\times Y//G'$.

This becomes a nice symmetric monoidal category; the identity in $Hom(G,G)$ is $T^*G$.

Suppose I want to consider the category of hyperkähler manifolds instead. I can try the following

  1. The objects are semisimple compact groups
  2. The morphisms $Hom(G,G')$ are hyperkähler manifolds with Hamiltonian $G\times G'$ action
  3. The composition of two morphisms $X\in Hom(G,G')$ and $Y\in Hom(G',G'')$ is given by the hyperkähler quotient $X\times Y///G'$.

Now, the problem is that $T^*G_\mathbb{C}$ has a hyperkähler metric (constructed by Kronheimer) and almost acts like an identity, but not quite: given a hyperkähler manifold $X$ with $G$ action, $T^*G_\mathbb{C} \times X /// G$ is equivalent to $G$ as holomorphic symplectic varieties but not equivalent as hyperkähler manifolds.

What should I do?

For my purpose, I guess using the terminology semigroupoid would suffice (I just want to define the target "category" of a TQFT precisely.) But I'm curious what kind of hell will break loose if I drop this axiom, why the people who originally defined categories included this into the axiom, etc.
show/hide this revision's text 2 correction

I was studying the axioms of a category, and noted that one axiom says there is an element $1_X\in Hom(X,X)$ for any object $X$ which serves as the identity. Why is this axiom necessary? What happens if I drop this axiom?

Background: I can define the category of affine holomorphic symplectic varieties, by saying

  1. The objects are semisimple algebraic groups
  2. The morphisms $Hom(G,G')$ are affine holomorphic symplectic varieties with Hamiltonian $G\times G'$ action
  3. The composition of two morphisms $X\in Hom(G,G')$ and $Y\in Hom(G',G'')$ is given by the holomorphic symplectic quotient $X\times Y//G$Y//G'$.

This becomes a nice symmetric monoidal category; the identity in $Hom(G,G)$ is $T^*G$.

Suppose I want to consider the category of hyperkähler manifolds instead. I can try the following

  1. The objects are semisimple compact groups
  2. The morphisms $Hom(G,G')$ are hyperkähler manifolds with Hamiltonian $G\times G'$ action
  3. The composition of two morphisms $X\in Hom(G,G')$ and $Y\in Hom(G',G'')$ is given by the hyperkähler quotient $X\times Y///G$Y///G'$.

Now, the problem is that $T^*G_\mathbb{C}$ has a hyperkähler metric (constructed by Kronheimer) and almost acts like an identity, but not quite: given a hyperkähler manifold $X$ with $G$ action, $T^*G_\mathbb{C} \times X /// G$ is equivalent to $G$ as holomorphic symplectic varieties but not equivalent as hyperkähler manifolds.

What should I do?
show/hide this revision's text 1

Why does Hom need an identity in the definition of the category?

I was studying the axioms of a category, and noted that one axiom says there is an element $1_X\in Hom(X,X)$ for any object $X$ which serves as the identity. Why is this axiom necessary? What happens if I drop this axiom?

Background: I can define the category of affine holomorphic symplectic varieties, by saying

  1. The objects are semisimple algebraic groups
  2. The morphisms $Hom(G,G')$ are affine holomorphic symplectic varieties with Hamiltonian $G\times G'$ action
  3. The composition of two morphisms $X\in Hom(G,G')$ and $Y\in Hom(G',G'')$ is given by the holomorphic symplectic quotient $X\times Y//G$.

This becomes a nice symmetric monoidal category; the identity in $Hom(G,G)$ is $T^*G$.

Suppose I want to consider the category of hyperkähler manifolds instead. I can try the following

  1. The objects are semisimple compact groups
  2. The morphisms $Hom(G,G')$ are hyperkähler manifolds with Hamiltonian $G\times G'$ action
  3. The composition of two morphisms $X\in Hom(G,G')$ and $Y\in Hom(G',G'')$ is given by the hyperkähler quotient $X\times Y///G$.

Now, the problem is that $T^*G_\mathbb{C}$ has a hyperkähler metric (constructed by Kronheimer) and almost acts like an identity, but not quite: given a hyperkähler manifold $X$ with $G$ action, $T^*G_\mathbb{C} \times X /// G$ is equivalent to $G$ as holomorphic symplectic varieties but not equivalent as hyperkähler manifolds.

What should I do?